In this paper we introduce and study graph analogues of Lebesgue and Hausdorff dimensions of topological spaces from the graph-theoretical point of view. We show that they are closely related to well-known graph characteristics such as rank dimension [1]} and Prague (or Nešetřil-Rödl) dimension [2]}. It
allowed us to define fractal graphs and determine fractality of some graph classes. In particular, it occurred that fractal graphs in some sense could be considered as generalizations of class 2 graphs. We demonstrate that various properties of dimensions of compact topological spaces have graph-theoretical analogues. Moreover, we also show how these relations allow for reverse transfer of combinatorial results to the general topology by proving a new property of general compact metric spaces using machinery from theory of hypergraphs. Finally, we show how Hausdorff (and Prague) dimension of graphs is related to Kolmogorov complexity. This relation allowed us to find a lower bound for Prague dimension for almost all graphs using incompressibility method from the theory of Kolmogorov complexity.
